Middle School Math Quiz: Solving For A Variable In Formulas

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Solving For A Variable In FormulasQuestion 1 of 7

The formula for the volume of a triangular prism is $V = \frac{1}{2}bhl$ , where $b$ is the base of the triangle, $h$ is the height of the triangle, and $l$ is the length of the prism. A triangular prism has a volume of $180$ cubic centimeters, a triangular base with height $6$ cm, and the prism length is $5$ cm. What is the length of the triangle's base?

8

10

12

15

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Middle School Math Quiz

Middle School Math Quiz: Solving For A Variable In Formulas

Practice Solving For A Variable In Formulas in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Solving For A Variable In Formulas, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$8$ cm
$10$ cm
$12$ cm (correct answer)
$15$ cm

Explanation: Substitute the known values:

180 = \frac{1}{2}b(6)(5)

. This simplifies to

180 = \frac{1}{2}b(30) = 15b

. Solving for b:

b = \frac{180}{15} = 12

cm. Choice A results from using the wrong formula

V = bhl

without the

\frac{1}{2}

. Choice B comes from confusing which variable represents which measurement. Choice D results from calculation errors when solving

15b = 180

Question 2

The formula $C = \frac{5}{9}(F - 32)$ converts temperature from Fahrenheit to Celsius. On a day when the temperature increased from $41°F$ to $68°F$ , what was the increase in Celsius degrees?

$12°C$
$15°C$ (correct answer)
$20°C$
$27°C$

Explanation: Convert both temperatures to Celsius. For 41°F:

C_1 = \frac{5}{9}(41 - 32) = \frac{5}{9}(9) = 5°C

. For 68°F:

C_2 = \frac{5}{9}(68 - 32) = \frac{5}{9}(36) = 20°C

. The increase is

20 - 5 = 15°C

. Choice A results from calculation errors in the conversions. Choice C gives the final Celsius temperature instead of the increase. Choice D gives the Fahrenheit increase instead of the Celsius increase.

Question 3

The formula $P = 2l + 2w$ gives the perimeter of a rectangle. A rectangular garden has a perimeter of $84$ feet and a length that is $6$ feet more than twice the width. What is the width of the garden?

$12$ feet (correct answer)
$14$ feet
$18$ feet
$20$ feet

Explanation: Let

w

be the width. Then the length is

l = 2w + 6

. Substitute into the perimeter formula:

84 = 2(2w + 6) + 2w = 4w + 12 + 2w = 6w + 12

. Solving:

84 - 12 = 6w

, so

72 = 6w

, thus

w = 12

feet. Choice B results from solving

84 = 2l + 2w

with

l = 2w + 6

but making arithmetic errors. Choice C gives the length instead of the width. Choice D results from setting up the equation incorrectly.

Question 4

The formula $d = rt$ relates distance, rate, and time. Two cars start from the same point and travel in opposite directions. Car A travels at $55$ mph and Car B travels at $65$ mph. After how many hours will they be $300$ miles apart?

$2.0$ hours
$2.5$ hours (correct answer)
$3.0$ hours
$4.6$ hours

Explanation: Since the cars travel in opposite directions, their combined rate of separation is

55 + 65 = 120

mph. Using

d = rt

300 = 120t

, so

t = \frac{300}{120} = 2.5

hours. Choice A results from using only one car's speed in the calculation. Choice C comes from arithmetic errors. Choice D results from incorrectly subtracting the speeds instead of adding them:

\frac{300}{65} \approx 4.6

Question 5

The formula $F = \frac{9}{5}C + 32$ converts Celsius to Fahrenheit. During a science experiment, the temperature needs to be maintained between $68°F$ and $86°F$ . What is the Celsius temperature range?

$15°C$ to $25°C$
$30°C$ to $40°C$
$25°C$ to $35°C$
$20°C$ to $30°C$ (correct answer)

Explanation: When you encounter temperature conversion problems with ranges, you need to convert both endpoints of the Fahrenheit range to find the corresponding Celsius range. This involves solving the conversion formula for C, then applying it to both boundary values. To find the Celsius equivalents, rearrange the formula

F = \frac{9}{5}C + 32

to solve for C. Subtract 32 from both sides:

F - 32 = \frac{9}{5}C

. Then multiply both sides by

\frac{5}{9}

C = \frac{5}{9}(F - 32)

. For the lower bound (68°F):

C = \frac{5}{9}(68 - 32) = \frac{5}{9}(36) = 20°C

For the upper bound (86°F):

C = \frac{5}{9}(86 - 32) = \frac{5}{9}(54) = 30°C

The Celsius range is 20°C to 30°C, making D correct. Looking at the wrong answers: A (15°C to 25°C) represents temperatures that are too cold - this range converts to roughly 59°F to 77°F. B (30°C to 40°C) gives temperatures that are too hot, converting to 86°F to 104°F. C (25°C to 35°C) is also too warm, converting to 77°F to 95°F. Study tip: When working with temperature conversion ranges, always solve the conversion formula for the unknown variable first, then substitute both endpoint values. Double-check by converting your answer back to the original units - your Celsius range should give you back the original Fahrenheit range.

Question 6

The formula $I = Prt$ calculates simple interest. If $2400 in interest is earned on a principal of $8000 over $ $5$ $ years, what is the annual interest rate as a percentage?

$4\%$
$10\%$
$8\%$
$6\%$ (correct answer)

Explanation: When you encounter a simple interest problem, you're working with the formula

I = Prt

, where

I

is interest earned,

P

is principal (starting amount),

r

is the annual interest rate (as a decimal), and

t

is time in years. Your goal is to solve for the unknown variable by substituting the given values. Here, you know

I = \$2400

P = \$8000

, and

t = 5

years. Substitute these into the formula:

2400 = 8000 \times r \times 5

. This simplifies to

2400 = 40000r

. Solving for

r

r = \frac{2400}{40000} = 0.06

. Since interest rates are expressed as percentages, multiply by 100:

0.06 \times 100 = 6\%

. Looking at the wrong answers: Choice A (

4\%

) would only generate

1600 in interest over 5 years (

8000 \times 0.04 \times 5

), which is too low. Choice B (

10\%

) would generate

4000 in interest, nearly double what was actually earned. Choice C (

8\%

) would produce $3200 in interest, also more than the given $2400. The key strategy for simple interest problems is to carefully substitute known values and solve algebraically for the unknown. Always remember to convert your final decimal answer to a percentage when the question asks for a rate. Double-check by plugging your answer back into the original formula to verify it produces the given interest amount.

Question 7

The formula $A = P(1 + rt)$ gives the amount $A$ after $t$ years when principal $P$ earns simple interest at rate $r$ . If Maria invests money for $6$ years at $4\%$ annual interest and the final amount is $1860, what was her initial investment?

$1200
$1500 (correct answer)
$1650
$1800

Explanation: Substitute known values:

1860 = P(1 + 0.04 \times 6)

. This gives

1860 = P(1 + 0.24) = P(1.24)

. Solving for P:

P = \frac{1860}{1.24} = 1500

. Choice A results from incorrectly using the compound interest formula or calculation errors. Choice C comes from solving

1860 = P + P(0.04)(6)

but making arithmetic mistakes. Choice D results from confusing the problem and thinking the interest earned was $60.